Emergent nearest-neighbor attraction in the fully renormalized interactions of the single-band repulsive Hubbard model at weak coupling

We compute the perturbative expansion for the effective interaction W of the half-filled 2-dimensional Hubbard model. We derive extensions of standard random-phase-approximation resummations that include arbitrarily high order contributions in the Wtt and Wt down arrow basis. Using algorithmic tools we explore the static Q-dependent interaction as well as the same-time quantity both in momentum and real space. We emphasize the absence of screening in the Hubbard interaction where we find an enhanced repulsive local Wt down arrow with a nonzero attractive Wtt. Finally, starting from only a locally repulsive bare interaction we find an emergent nonlocal nearest-neighbor attraction for low temperatures at sufficiently large values of U/t which may be key to understanding pairing processes in the model.